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# Stable phase retrieval and perturbations of frames

Dec 2022

A frame $(x_j)_{j\in J}$ for a Hilbert space $H$ is said to do phaseretrieval if for all distinct vectors $x,y\in H$ the magnitude of the framecoefficients $(|\langle x, x_j\rangle|)_{j\in J}$ and $(|\langle y,x_j\rangle|)_{j\in J}$ distinguish $x$ from $y$ (up to a unimodular scalar). Aframe which does phase retrieval is said to do $C$-stable phase retrieval ifthe recovery of any vector $x\in H$ from the magnitude of the framecoefficients is $C$-Lipschitz. It is known that if a frame does stable phaseretrieval then any sufficiently small perturbation of the frame vectors will dostable phase retrieval, though with a slightly worse stability constant. Weprovide new quantitative bounds on how the stability constant for phaseretrieval is affected by a small perturbation of the frame vectors. Thesebounds are significant in that they are independent of the dimension of theHilbert space and the number of vectors in the frame.

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